A circle \(S\equiv x^2+y^2+4x+2fy+c=0\) passes through the centre of the circle \(x^2+y^2-4x+6y-2=0\). If the line \(3x-3y=c\) passes through the centre of the circle \(S=0\), then the length of tangent from \((1,1)\) to the circle \(S=0\) is
Show Hint
- \(8\)
- \(\sqrt{2}\)
- \(5\)
- \(\sqrt{10}\)
The Correct Option is C
Solution and Explanation
\[ x^2+y^2-4x+6y-2=0 \] Center: \[ (2,-3) \] Since this lies on \(S\): \[ 4+9+8-6f+c=0 \] \[ 21-6f+c=0 \] \[ c=6f-21 \]
Step 2: Use center condition.
Center of \(S\): \[ (-2,-f) \] Given: \[ 3x-3y=c \] Substitute: \[ 3(-2)-3(-f)=c \] \[ -6+3f=c \] Equating: \[ 3f-6=6f-21 \] \[ 15=3f \] \[ f=5 \] Thus: \[ c=9 \]
Step 3: Find tangent length.
Circle: \[ x^2+y^2+4x+10y+9=0 \] At \((1,1)\): \[ 1+1+4+10+9=25 \] Length: \[ =\sqrt{25}=5 \]
Top TS EAMCET Mathematics Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is
Top TS EAMCET Coordinate Geometry Questions
A random variable X has the following probability distribution
X= x 1 2 3 4 5 6 7 8 P(X = x) 0.15 0.23 k 0.10 0.20 0.08 0.07 0.05 For the events E = {x/x is a prime number} and F = {x/x <4} then P(E ∪ F)
5 persons entered a lift cabin in the cellar of a 7-floor building apart from cellar. If each of the independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is
If a point P moves so that the distance from (0,2) to P is \(\frac{1}{√2 }\) times the distance of P from (-1,0), then the locus of the point P is
Let d be the distance between the parallel lines 3x - 2y + 5 = 0 and 3x - 2y + 5 + 2√13 = 0. Let L1 = 3x - 2y + k1 = 0 (k1 > 0) and L2 = 3x - 2y + k2 = 0 (k2 > 0) be two lines that are at the distance of \(\frac{4d}{√13}\) and \(\frac{3d}{√13}\) from the line 3x - 2y + 5y = 0. Then the combined equation of the lines L1 = 0 and L2 = 0 is:
If (h,k) is the image of the point (3,4) with respect to the line 2x - 3y -5 = 0 and (l,m) is the foot of the perpendicular from (h,k) on the line 3x + 2y + 12 = 0, then lh + mk + 1 = 2x - 3y - 5 = 0.
Top TS EAMCET Questions
- A bag contains $5$ red balls, $3$ black balls and $4$ white balls are drawn at random. The probability that they are not of same colour is
- If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta $ lies is
- If $a$ is a unit vector, then $| a \times \hat{ i }|^{2}+| a \times \hat{ j }|^{2}+| a \times \hat{ k }|^{2}=$
- $A$ straight line makes an intercept on the $Y$-axis twice as long as that on $X$-axis and is at a unit distance from the origin. Then the line is represented by the equations
- Let $S$ and $S'$ be the foci of an ellipse and B be one end of its minor axis. If $SBS'$ is an isosceles right angled triangle then the eccentricity of the ellipse is