Question:
If the line \(4x+3y = 7\) touches the hyperbola \(x^2-y^2 = 7\), then the sum of the co-ordinates of the point of contact is...
If the line \(4x+3y = 7\) touches the hyperbola \(x^2-y^2 = 7\), then the sum of the co-ordinates of the point of contact is...
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Compare the line with the tangent form x x1 - y y1 = 7.
Updated On: Oct 1, 2026
- \(4\)
- \(7\)
- \(1\)
- \(0\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
The tangent to \(x^2-y^2=7\) at the point \((x_1,y_1)\) on it is \(xx_1-yy_1=7\). If the given line touches the hyperbola, it must be this same line.
Step 2: Compare:
Write \(4x+3y=7\). Match with \(x_1x-y_1y=7\):
\[ x_1=4,\qquad -y_1=3\ \Rightarrow\ y_1=-3 \]
Step 3: Check the point lies on the hyperbola:
\(x_1^2-y_1^2=16-9=7\). Yes.
Step 4: Sum of coordinates:
\(4+(-3)=1\), so the answer is (C).
Final Answer:
The point of contact is (4, -3), sum 1. \[ \boxed{1} \]
The tangent to \(x^2-y^2=7\) at the point \((x_1,y_1)\) on it is \(xx_1-yy_1=7\). If the given line touches the hyperbola, it must be this same line.
Step 2: Compare:
Write \(4x+3y=7\). Match with \(x_1x-y_1y=7\):
\[ x_1=4,\qquad -y_1=3\ \Rightarrow\ y_1=-3 \]
Step 3: Check the point lies on the hyperbola:
\(x_1^2-y_1^2=16-9=7\). Yes.
Step 4: Sum of coordinates:
\(4+(-3)=1\), so the answer is (C).
Final Answer:
The point of contact is (4, -3), sum 1. \[ \boxed{1} \]
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