Question

Suppose \( d_1 \) and \( d_2 \) are respectively the lengths of intercepts of the circle \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 10x - 14y + 65 = 0 \) on the line \( 2x - 2y = 3 \). Then which of the following is true?

• A. \( d_1 = 2d_2 \)
• B. \( d_2 = 2d_1 \)
• C. \( d_1 = 3d_2 \)
• D. \( d_1 = d_2 \)

Hint:

Length of Intercept on Circle}
Use \( \text{Length} = 2\sqrt{r^2 - d^2} \)
Always convert general form of circle to standard form
Use distance formula carefully when dealing with line in general form

Answer 1

The Correct Option is D

Let’s write the line as: \[ 2x - 2y = 3 \Rightarrow x - y = \frac{3}{2} \] Circle 1: \( x^2 + y^2 = 4 \Rightarrow \) Center \( (0, 0) \), Radius = 2 Circle 2: Rewrite in standard form: \[ x^2 + y^2 - 10x - 14y + 65 = 0 \Rightarrow (x - 5)^2 + (y - 7)^2 = 25 + 49 - 65 = 9 \Rightarrow \text{Center } (5, 7), \text{ Radius } = 3 \] Length of intercept a line makes on a circle: \[ \text{Length} = 2\sqrt{r^2 - d^2} \] Where \( d \) is the perpendicular distance from center to line Compute \( d_1 \) for circle 1: \[ \text{Distance from } (0, 0) \text{ to } x - y = \frac{3}{2} = \frac{3/2}{\sqrt{2}} = \frac{3}{2\sqrt{2}} \] \[ d_1 = 2\sqrt{4 - \left( \frac{9}{8} \right)} = 2\sqrt{\frac{32 - 9}{8}} = 2\sqrt{\frac{23}{8}} \] Do same for circle 2: Distance from (5,7) to \( x - y = \frac{3}{2} \) is: \[ |5 - 7 - \frac{3}{2}| = \left| -2 - \frac{3}{2} \right| = \frac{7}{2}, \text{ Denominator: } \sqrt{2} \Rightarrow d_2 = \frac{7}{2\sqrt{2}}, \text{ similar value } \] After simplification both yield the same result.