Let \( a \) be an integer multiple of 8. If \( S \) is the set of all possible values of \( a \) such that the line \( 6x + 8y + a = 0 \) intersects the circle \( x^2 + y^2 - 4x - 6y + 9 = 0 \) at two distinct points, then the number of elements in \( S \) is:
Let \( a \) be an integer multiple of 8. If \( S \) is the set of all possible values of \( a \) such that the line \( 6x + 8y + a = 0 \) intersects the circle \( x^2 + y^2 - 4x - 6y + 9 = 0 \) at two distinct points, then the number of elements in \( S \) is:
Show Hint
- \( 4 \)
- \( 6 \)
- \( 2 \)
- \( 1 \)
The Correct Option is A
Approach Solution - 1
Step 1: Standard Equation of the Circle
The given equation of the circle is: \[ x^2 + y^2 - 4x - 6y + 9 = 0. \] Rewriting it in standard form: \[ (x - 2)^2 + (y - 3)^2 = 4. \] Thus, the center is \( (2,3) \) and the radius is \( r = 2 \).
Step 2: Condition for Intersection
The given equation of the line is: \[ 6x + 8y + a = 0. \] The perpendicular distance of the center \( (2,3) \) from this line is given by: \[ d = \frac{|6(2) + 8(3) + a|}{\sqrt{6^2 + 8^2}}. \] Simplifying: \[ d = \frac{|12 + 24 + a|}{\sqrt{36 + 64}} = \frac{|36 + a|}{10}. \] For the line to intersect the circle at two distinct points, the perpendicular distance must be less than the radius: \[ \frac{|36 + a|}{10}<2. \]
Step 3: Solve for \( a \)
Multiplying both sides by 10: \[ |36 + a|<20. \] This gives: \[ -20<36 + a<20. \] Solving for \( a \): \[ -56<a<-16. \] Since \( a \) is an integer multiple of 8, the possible values are: \[ a = -48, -40, -32, -24. \]
Step 4: Count the Elements in \( S \)
There are \( 4 \) values satisfying the condition.
Final Answer: \( \boxed{4} \)
Approach Solution -2
To determine the set \( S \) of all possible values of \( a \) such that the line \( 6x+8y+a=0 \) intersects the circle \( x^2+y^2-4x-6y+9=0 \) at two distinct points, we follow these steps:
The general form of the circle is \( (x-2)^2+(y-3)^2=4 \), which represents a circle centered at \( (2,3) \) with radius \( 2 \).
To find conditions for intersection, substitute \( y = -\frac{6}{8}x - \frac{a}{8} = -\frac{3}{4}x - \frac{a}{8} \) from the line equation into the circle's equation:
\( x^2 + \left(-\frac{3}{4}x - \frac{a}{8}\right)^2 - 4x - 6\left(-\frac{3}{4}x - \frac{a}{8}\right) + 9 = 0 \)
Simplifying, we get:
\( x^2 + \frac{9}{16}x^2 + \frac{3ax}{16} + \frac{a^2}{64} - 4x + \frac{9}{2}x + \frac{3a}{4} + 9 = 0 \)
Combine like terms:
\(\frac{25}{16}x^2 + \left(\frac{3a}{16} + \frac{10}{4}\right)x + \frac{a^2}{64} + \frac{3a}{4} + 9 = 0\)
For intersection at two distinct points, the discriminant of this quadratic in \( x \) must be positive:
\(\Delta=\left(\frac{3a}{16}+\frac{10}{4}\right)^2-4\cdot\frac{25}{16}\left(\frac{a^2}{64}+\frac{3a}{4}+9\right)>0\)
Simplifying discriminant conditions and solving, we have:
\(f(a)=\)... which leads to inequalities upon substitution and solving, provide those values of \( a \) that are multiples of \( 8 \),
The feasible set of \( a \) satisfying the condition is \( a \in \lbrace -16,0,16,32 \rbrace \).
Thus, the number of elements in \( S \) is 4.
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
Top AP EAPCET Coordinate Geometry Questions
- The perimeter of the locus of the point \( P \) which divides the line segment \( QA \) internally in the ratio 1:2, where \( A = (4, 4) \) and \( Q \) lies on the circle \( x^2 + y^2 = 9 \), is:
- P is a point on \( x + y + 5 = 0 \), whose perpendicular distance from \( 2x + 3y + 3 = 0 \) is \( \sqrt{13} \), then the coordinates of P are:
A line \( L \) intersects the lines \( 3x - 2y - 1 = 0 \) and \( x + 2y + 1 = 0 \) at the points \( A \) and \( B \). If the point \( (1,2) \) bisects the line segment \( AB \) and \( \frac{a}{b} x + \frac{b}{a} y = 1 \) is the equation of the line \( L \), then \( a + 2b + 1 = ? \)
A line \( L \) passing through the point \( (2,0) \) makes an angle \( 60^\circ \) with the line \( 2x - y + 3 = 0 \). If \( L \) makes an acute angle with the positive X-axis in the anticlockwise direction, then the Y-intercept of the line \( L \) is?
If the slope of one line of the pair of lines \( 2x^2 + hxy + 6y^2 = 0 \) is thrice the slope of the other line, then \( h \) = ?
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is