If the line $3x-4y=m$ intersects the circle $x^{2}+y^{2}=4x+8y+5$ in two distinct points, then :

Question

Accepted Answer

To determine why Option 2 ($-35 < m < 15$) is the correct answer, we need to analyze the conditions under which the line $3x - 4y = m$ intersects the circle $x^2 + y^2 = 4x + 8y + 5$ in two distinct points. ### Step-by-Step Explanation: 1. **Equation of the Circle:** The given circle equation is $x^2 + y^2 = 4x + 8y + 5$. We can rewrite it in standard form by completing the square: $$ x^2 - 4x + y^2 - 8y = -5 $$ Completing the square for $x$ and $y$: $$ (x - 2)^2 - 4 + (y - 4)^2 - 16 = -5 $$ $$ (x - 2)^2 + (y - 4)^2 = 25 $$ This represents a circle with center $(2, 4)$ and radius $5$. 2. **Equation of the Line:** The line is given by $3x - 4y = m$. We can express $y$ in terms of $x$: $$ y = \frac{3}{4}x - \frac{m}{4} $$ 3. **Intersection Condition:** For the line to intersect the circle in two distinct points, the quadratic equation formed by substituting $y = \frac{3}{4}x - \frac{m}{4}$ into the circle's equation must have two distinct real roots. This requires the discriminant of the resulting quadratic equation to be positive. 4. **Substitute and Simplify:** Substitute $y = \frac{3}{4}x - \frac{m}{4}$ into $(x - 2)^2 + (y - 4)^2 = 25$: $$ (x - 2)^2 + \left(\frac{3}{4}x - \frac{m}{4} - 4\right)^2 = 25 $$ Simplify and expand this equation to form a quadratic in $x$. The discriminant $\Delta$ of this quadratic must be greater than zero for two distinct intersection points. 5. **Discriminant Analysis:** After simplification, the discriminant $\Delta$ will be a function of $m$. Solving $\Delta > 0$ will give the range of $m$ for which the line intersects the circle at two distinct points. 6. **Conclusion:** Solving the inequality $\Delta > 0$ yields the range $-35 < m < 15$. This is why Option 2 is correct. ### Why Other Options are Incorrect: - **Option 1 ($-85 < m < -35$):** This range does not satisfy the condition $\Delta > 0$ for the quadratic formed by the intersection of the line and the circle. The line would either not intersect the circle or be tangent to it. - **Option 3 ($15 < m < 65$):** Similarly, this range does not satisfy the condition for two distinct intersection points. The line would either not intersect the circle or be tangent to it. - **Option 4 ($35 < m < 85$):** This range also fails to satisfy the condition for two distinct intersection points. The line would either not intersect the circle or be tangent to it. Thus, the only range that allows the line to intersect the circle at two distinct points is $-35 < m < 15$, confirming that Option 2 is indeed the correct answer.